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Geometric flows are ubiquitous in modern geometry, topology, and theoretical physics. Intuitively, a geometric flow is a rule for deforming a manifold according to its curvature at each point. These flows come in two flavors: intrinsic flows, where the flow is of an abstract Riemannian manifold, and extrinsic flows, where the flow is of a surface immersed in an ambient Riemannian Manifold. An example of the former type is Ricci Flow, famous for its use in Perelman's proof of the Poincare Conjecture. An example of the latter type is Mean Curvature Flow, which both models a number of physical systems and has recently been used to reveal a great deal about the geometry and topology of surfaces in Euclidean Space. Additionally, Inverse Mean Curvature Flow, a closely-related extrinsic flow, has garnered increasing interest in recent years thanks to its applications to mathematical physics.

I will begin this talk with some background on extrinsic geometry, then turn my attention to Mean Curvature Flow. In particular, I will explain some well-known results about singularities of Mean Curvature Flow. With this serving as a reference point, I will then speak on Inverse Mean Curvature Flow, and end by presenting some of my own results on its singularities for comparison. I intend to make this talk accessible to non-experts in differential geometry.